How to select and merge vertices into separate triangles?
Switch to edge select mode (CtrlTab&gt; Edge)

Select the edges you want to cut. Border select works well (B), or you can select a couple edges like the ones you want to select and press ShiftG&gt; Face Angles.

Press W&gt; Subdivide and turn up the smoothness in the redo panel.



Note
I very much recommend working with quads instead, and using a triangulate modifier to non-destructively triangulate the mesh in realtime.